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arXiv · 2303.10530

Tur\'an density of long tight cycle minus one hyperedge

József Balogh·Haoran Luo
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Abstract

Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Tur\'an density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Tur\'an density of $\mathcal{C}^-_{\ell}$ is $1/4$, for every sufficiently large $\ell$ not divisible by $3$. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidick\'y.

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József Balogh, Haoran Luo. 2023-03-19. Tur\'an density of long tight cycle minus one hyperedge. https://arxiv.org/abs/2303.10530

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